The essential musical theme that is the "melody" of a fugue.

The subject is presented at the beginning of every fugue, usually in an unaccompanied solo voice. Most fugues begin with a full exposition, where each voice sings the subject, one after another in turn, enabling the listener to clearly grasp and remember the subject. The subject is the musical kernel, the seed of the fugue; the fugue is very much "all about" the subject. Throughout the fugue, the subject comes and goes, appearing in different voices at different times. There are almost always episodes where the subject is absent though even the content of the episode is often derived from the subject. Hearing the subject play hide and seek it the first order narrative principle of a fugue. Equally important is the constantly shifting context in which the subject appears due to the affect of other voices in counterpoint. There are numerous ways that the subject may be modified during the fugue (inversion, augmentation, diminution). Occasionally there are "false" entries where only an initial fragment of the subject is heard; this is another aspect of the game. Fugues can even have multiple subjects (double fugue).

Subjects vary tremendously in all aspects: melody, harmonic implications, rhythm, length and complexity. Yet each and every subject seems to be complete, to posses its own self-sufficient musical integrity. Bach's fugues are valued, among other things, for the sheer artistry of his subjects. Given the importance of the subject in determining the entire nature of the fugue, it is logical that Bach's fugues, by definition, are profound because his subjects are.

From Bruhn:

“It has often been observed that it would seem wrong to say a fugue has a subject; one should rather say there is a subject which has generated a fugue. The perfect little musical entity which we call subject is in fact at the origin of the fugue. Its “companions” are dependent on it to the largest imaginable extent: were the subject any different, they too would not be what they are. The subject is responsible for the feelings of density and relaxation in the fugue and is the main force in creating structure. Whenever it rests for a while, its absence is distinctly felt.”

Because the subject is so crucial to the nature of the entire fugue, it is interesting to compare and contrast the subjects in the Well-Tempered Clavier. You can look, listen and navigate using the subject tables:

Bach - Well-Tempered Clavier - Book 1 Subject Table
Bach - Well-Tempered Clavier - Book 2 Subject Table

You can also study the subjects from Shostakovich's 20th century fugues:

Shostakovich - Op. 87 Subject Table

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